Finding the Sum of a Decimal Expression

We need to find the sum of the expression: \(\frac{0.01}{0.1}+\frac{0.1}{0.01}\). This involves calculating the value of each fraction and then adding them together.

Evaluating the First Term: \(\frac{0.01}{0.1}\)

To evaluate the first term, we can remove the decimals by multiplying both the numerator and the denominator by a power of 10. The highest number of decimal places is two (in 0.01), so we multiply by \(10^2 = 100\).

$$ \frac{0.01}{0.1} = \frac{0.01 \times 100}{0.1 \times 100} = \frac{1}{10} $$

As a decimal, \(\frac{1}{10}\) is equal to \(0.1\).

Evaluating the Second Term: \(\frac{0.1}{0.01}\)

Similarly, to evaluate the second term, we multiply both the numerator and the denominator by 100 to remove decimals.

$$ \frac{0.1}{0.01} = \frac{0.1 \times 100}{0.01 \times 100} = \frac{10}{1} $$

As a decimal, \(\frac{10}{1}\) is equal to \(10\).

Calculating the Sum

Now we add the values of the two terms we calculated:

$$ \text{Sum} = \text{First Term} + \text{Second Term} $$

$$ \text{Sum} = 0.1 + 10 $$

$$ \text{Sum} = 10.1 $$

The sum is \(10.1\). Looking at the options, \(10.1\) is the same as \(10.10\).

Comparing with Options

Let's check the calculated sum against the given options:

• Option 1: 10.11
• Option 2: 10.10
• Option 3: 1.10
• Option 4: 11.10

Our calculated sum, \(10.1\), matches Option 2, which is \(10.10\).

Therefore, the sum of \(\frac{0.01}{0.1}+\frac{0.1}{0.01}\) is \(10.10\).